Decimal 2 Binary Transforming Numerical Systems

Lesson 3: Decimal 2 Binary

Number we want to transform uses base X: we would like to transform this number into new one, that uses Y base. In this case we will use method of continuous dividing and multiplying (dividing numbers left of comma (,) with Y and multiplying numbers on the right side of comma, by Y - rational numbers). This method is best understood looking at these examples where X equals 10 (decimal base, B=10), and Y eqauls 2 (binary base, B=2)


Decimal System:

Base: 10
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9


Binary System:

Base: 2
Digits: 0, 1
Example: 27 (decimal) = 11011 ₂ (binary)




Decimal 2 Binary



Example

Transform 29 (decimal) into binary number - reading upwards, result is 11101₂

29 : 2 = 14 , remaining 1 20 last (smallest) digit
14 : 2 = 7 , remaining 0 21
 7 : 2 = 3 , remaining 1 22
 3 : 2 = 1 , remaining 1 23 
 1 : 2 = 0 , remaining 1 24 first digit

- end of procedure





Example

Transform 0,8125 (decimal) into binary number – reading downwards, result is 0,1101₂

0,8125 * 2 = 1, 625 2-1 first digit after zero
0,625 * 2 = 1, 25 2-2
0,25 * 2 = 0, 5 2-3
0,5 * 2 = 1, 0 2-4 last digit

0,0 end of procedure

Example

Transform 0,3 (decimal) into binary number – result is 0,01001 1001 1001… ₂shortly rounded = 0,01001₂

0,3 * 2 = 0, 6 2-1 first digit after zero
0,6 * 2 = 1, 2 2-2
--------
0,2 * 2 = 0, 4 2-3
0,4 * 2 = 0, 8 2-4
0,8 * 2 = 1, 6 2-5
0,6 * 2 = 1, 2 2-6
--------
…
…

procedure never ends

Example – Quick Method

Transform 53(decimal) into binary number using quick method

 53 25 + 24 + 22 + 20 =
- 32 -> 25 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20 = 
-------- 110101 2 
 21
- 16 -> 24 
--------
 5
- 4 -> 22 
--------
 1
- 1 -> 20 
--------
 0 -> end of procedure